## Multiple Equilibria

Another major task in formulating a model of metabolism is the inclusion of quantitative descriptions of the concentrations of complexes between metal cations and, primarily, phosphorylated metabolic intermediates. The procedure used to calculate these concentrations turns out to be formally equivalent to that used to determine the values of unitary rate constants, given a set of steady-state kinetic parameters. In the present case the initial concentrations of all reactants must be specified, together with expressions for, and values of, the various equilibrium constants of the binding reactions. Having made this claim it is probably most convincing to simply illustrate the process of analysis; it is as follows.

Q: Calculate the concentrations of free Mg, free ATP, and their 1:1 complex in a reaction mixture that has attained equilibrium, from a total concentration of 3 mmol L-1 Mg and 1 mmol L-1 ATP. Assume that the reaction scheme is

Mg2+ + ATP4- MgATP2-, KMgATP = 1 x 102 M-1 . [3.23]

A: We begin by writing the expression for the equilibrium constant of the reaction; it is eq% : = Kmgatp =

mgatp2mi

mg x atp4mi

The following conservation of mass conditions also apply.

eqn2 : = mg + mgatp2mi == 3.0 x 10-3 ; eqn3 : = atp4mi + mgatp2mi == 1.0 x 10-3 ;

Thus there exist three equations in three unknowns which we solve using the function Solve.

solution= Solve[{eqn1, eqn2, eqn3}, {mg, atp4mi, mgatp2mi}]

{{mgatp2mi^ 0.00021767, mg^ 0.00278233, atp4mi^ 0.00078233}, {mgatp2mi ^ 0.0137823, mg ^-0.0107823, atp4mi 0.0127823}}

The output is a list of two solutions but we note by inspection that only one of them is physically meaningful because all concentrations must be positive. The only meaningful solution shows that the concentration of free Mg is 2.782 mM, and that of its 1:1 complex with ATP is 0.218 mM. A simple way to delete the nonphysical solution is with the following input which relies on a number of Mathematica pattern recognition commands.

realSolution = solution / .

({a_, b_ -> c_, d_} /; Negative[c]) -> {} // Flatten

{mgatp2mi ^ 0.00021767, mg^ 0.00278233, atp4mi ^ 0.00078233}

It is left to the reader to use the Mathematica help browser to understand these commands and syntax of this input.

Also by way of a check on the solution to the problem we note that the conservation of mass condition for total ATP is satisfied since 2.782 mM + 0.218 mM = 3.0 mM. The test for conservation of mass is always worth carrying out to ensure your program is functioning correctly. This can be tested with the following input:

({a_, b_ -> c_, d_}/; replacement rule which converts a list of rules

Negativefc]) -> {} containing negative values on the rhs toanemptyset list // Flatten removes the inner set of braces from the list

Removing non-physically meaningful solutions from a list of replacement rules.

Q: Consider the more realistic reaction scheme than that given above in which ATP exists in various protonated forms and these have different values of the binding constant for Mg and Ca. In addition, suppose that Ca is present in the solution as well. Calculate the concentrations of free Mg, Ca, and the various protonated forms of ATP and their complexes in a reaction mixture that has attained equilibrium from a total concentration of each species of 3 mmol L-1 . Assume that the reaction scheme is as follows, noting that here we use the superscripted valences of the ions to help emphasize the net charges of the different complexes.

The complexes of Mg and ATP are formed in the following reactions with their respective equilibrium constants.

Mg2+ + ATP4- h-i MgAT P2-, KMgATP2- = 1 x 102M-1 , [3.24]

Mg2+ + ATP3- h ~ I Mg ATP-, Kmhatp- = 7 x 103M-1 . [3.25]

Complexes of Ca and ATP are

Ca2+ + ATP4- ^ CaATP2-, KCiiATP2- = 1 x 102 M-1 , [3.26]

Ca2+ + ATP3- HOOAS CaATP-, K^atp- = 1 x 104 M-1 . [3.27]

Protonation of ATP also occurs.

H+ + ATP4- HOATOi ATP3-, KATP3- = 3 x 106 M-1 , [3.28]

H+ + ATP3- hAOOi ATP2-, KATP2- = 1 x 104 M-1 . [3.29]

A: We solve this problem in a manner similar to that used in the previous question, by setting up the expressions for the equilibrium constants and defining the conservation of mass conditions.

The equilibrium equations for the reactions in Eqns [3.24 - 3.29] are mgatp2mi 2

mg x atp4mi mgatpmi 3

eqn2 : = Kmgatpmi == " ~ ; Kmgatpmi = 7 x 10 ;

mg x atp3mi caatp2mi 2

eqn3 := Kcaatp2mi == ---"" ; Kcaatp2mi = 1 x 10 ;

ca x atp4mi caatpmi 4

eqn4 : = Kcaatpmi == -"—"" ; Kcaatpmi = 1 x 10 ;

ca x atp3mi atp3mi 6

atp4mi atp2mi 4

eqn6 := Katp2mi x h ==---—-- ; Katp2mi = 1 x 10 ;

atp3mi

The conservation of mass equations are eqn7 : = mg + mgatp2mi + mgatpmi == 3.0 x 10-3 ; eqn8 : = ca + caatp2mi + caatpmi == 3.0 x 10-3 ; eqn9 : = atp4mi + atp3mi + atp2mi +

caatp2mi + caatpmi + mgatp2mi + mgatpmi == 5.0 x 10-3 ;

Hence there are 9 equations and 10 unknowns. By setting the pH to 7.2 and then calculating the H+ concentration, we can solve for the remaining 9 unknowns.

unknowns = {mg, ca, atp4mi, atp3mi, atp2mi, mgatp2mi, mgatpmi, caatp2mi, caatpmi};

solution = Solve[{eqn1, eqn2 , eqn3 , eqn4 , eqn5 , eqn6 , eqn7 , eqn8 , eqn9 } , unknowns];

realsolution = solution / .

({a_, b_ -> c_, d_} /; Negative[c]) -> {} // Flatten;

MatrixForm[Transpose[{unknowns, unknowns /. realsolution}]]

 mg 0 .00122309 ca 0 000989526 atp4mi 0 .00101951 atp3mi 0 .00019298 atp2mi 1 21762 x 10-7 mgatp2mi 0 000124695 mgatpmi 0 .00165222 caatp2mi 0 000100883 caatpmi 0 .00190959

Note that we have used the new commands MatrixForm and Transpose to display the solutions as a matrix.

MatrixFormffof] diplays the list in matrix form Transposefm] Transposes the matrix m

### Displaying lists.

Papers by Conigrave and Morris(1) and Mulquiney and Kuchel(2'3) treat even more complicated reaction schemes than those that are considered here, but the basic principles of solving the nonlinear algebraic equations to yield estimates of all the concentrations of the reactants are exactly the same. The nonlinearity of the system comes about via the definitions of the equilibrium constants, since the expressions are ratios of products of the equilibrium concentrations of the reactants. The other important aspect of setting up the analysis is the definition of the conservation of mass conditions. In these expressions special care is needed when taking into account the stoichiometry of the complexes.